Parameter-free implementation of the quadratic $C^0$ interior penalty method for the biharmonic equation

The symmetric $C^0$ interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic selection of the involved stability parameter in terms of the geometry of the underlying triangulation. It avoids any extra input variables and guarantees a stable discretization with stability $1/2$. The practical realization of this parameter-free scheme in less than 120 lines of MATLAB is described and explained in detail. Numerical experiments show stability constants smaller than 1 and so over-stabilization is excluded. The approach is documented for the lowest-order scheme in 2D for triangles, but the methodology behind can be generalized for automatic local parameter decisions.